"They cannot falsely accuse any of our people and we just let it pass," Mr Museveni said in the May 11 letter.
President Museveni has directed Foreign Affairs minister Jeje Odongo to write to demand the UK government explain the source their information that Speaker Anita Among has a house or houses in the UK.
"They cannot falsely accuse any of our people and we just let it pass," Mr Museveni said in the May 11 letter.
The President said he had received the Inspectorate of Government's message that Ms Among did not not reveal anything related to ownership of a house or houses in the UK.
"I have received a letter from Dr Patricia Achan Okiria, deputy Inspector General of Government, of the 9th of May, 2024, informing me that the Rt. Hon. Anita Annet Among never revealed that she had a house in United Kingdom," the President told Gen Odongo.
He said Ms Among had also told him that she does not own a house or houses in UK.
"Therefore, working with the Attorney-General, write an appropriate letter to the relevant authorities in UK to demand the source of their information," the President added.
Speaker Among's alleged UK wealth was revealed to President Museveni by the British High Commissioner, Ms Kate Airey OBE, at State House Nakasero on April 30 - just hours before the UK government announced sanctions against three Ugandan legislators.
Ms Airey was meeting the President to brief him about the impending sanctions.
Sign up for free AllAfrica Newsletters
Get the latest in African news delivered straight to your inbox
"She told me that they were going to sanction Speaker Among. I said: 'why?'. She said Anita Among has got a house or houses in UK and has got bank accounts from which she pays schools fees for her children who are studying there," the President wrote.
"I told her that the issue of houses would be very interesting if, especially, Anita Among did not declare them in her Leadership Code documents," he added.
The President said if Ms Among indeed had houses in the UK, then the next issue would be to ascertain where she got the money to build them.
On May 2, the President directed IGG and Ethics ministry to look into Speaker's UK wealth.
"Why would a Ugandan leader build or buy houses in UK or anywhere else abroad, when Uganda, the still under-developed country where she would have earned the money, still needs those investments?" the President asked.
"It would definitely be a moral and ideological mistake."
The UK government slapped Speaker Among and two former Karamoja Affairs ministers Mary Goretti Kitutu and Agnes Nandutu with travel ban and asset freezes for what it called "epitome of corruption and [that has] no place in society".
While Kitutu and Nandutu are facing prosecution, Ms Among swiftly returned 500 pieces of iron sheets she had received when the revelations came out.
The iron sheets were from a government-funded housing project aimed at assisting vulnerable communities in Karamoja.
Speaker Among was found to have benefited from the proceeds of the theft, according to the statement released by London.
